Jamila sold a table and a chair for $Rs.\, 1050,$ thereby making a profit of $10\%$ on the table and $25\%$ on the chair. If she had taken a profit of $25\%$ on the table and $10\%$ on the chair,she would have got $Rs.\, 1065.$ Find the cost price of each (in $Rs.$).

In a competitive examination,one mark is awarded for each correct answer while $\frac{1}{2}$ mark is deducted for every wrong answer. Jayanti answered $120$ questions and got $90$ marks. How many questions did she answer correctly?

Vijay had some bananas,and he divided them into two lots $A$ and $B$. He sold the first lot at the rate of $Rs. 2$ for $3$ bananas and the second lot at the rate of $Rs. 1$ per banana,and got a total of $Rs. 400$. If he had sold the first lot at the rate of $Rs. 1$ per banana,and the second lot at the rate of $Rs. 4$ for $5$ bananas,his total collection would have been $Rs. 460$. Find the total number of bananas he had.

Is it true to say that the pair of equations $-x + 2y + 2 = 0$ and $\frac{1}{2}x - \frac{1}{4}y - 1 = 0$ has a unique solution? Justify your answer.

Solve the following pair of linear equations:
$152x - 378y = -74$
$-378x + 152y = -604$

If $2x + y = 23$ and $4x - y = 19$,find the values of $5y - 2x$ and $\frac{y}{x} - 2$.